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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,8]))
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pari:[g,chi] = znchar(Mod(737,740))
\(\chi_{740}(33,\cdot)\)
\(\chi_{740}(53,\cdot)\)
\(\chi_{740}(157,\cdot)\)
\(\chi_{740}(197,\cdot)\)
\(\chi_{740}(293,\cdot)\)
\(\chi_{740}(377,\cdot)\)
\(\chi_{740}(453,\cdot)\)
\(\chi_{740}(477,\cdot)\)
\(\chi_{740}(493,\cdot)\)
\(\chi_{740}(497,\cdot)\)
\(\chi_{740}(673,\cdot)\)
\(\chi_{740}(737,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((371,297,261)\) → \((1,i,e\left(\frac{2}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 740 }(737, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
